Determinants vs. Algebraic Branching Programs

TL;DR

This paper establishes a close relationship between determinantal complexity and algebraic branching program (ABP) complexity for homogeneous polynomials, showing that bounds on one imply bounds on the other, thus connecting two major open problems in algebraic complexity.

Contribution

It proves that for homogeneous polynomials, low determinantal complexity implies low ABP complexity, and for most polynomials, the ABP width is tightly bounded, linking two key complexity measures.

Findings

Homogeneous polynomials with determinantal complexity s can be computed by ABPs of size O(d^5 s)

For most homogeneous polynomials, the ABP width is just s-1 with size O(ds)

Super-linear lower bounds for ABPs imply similar bounds for determinantal complexity

Abstract

We show that for every homogeneous polynomial of degree $d$, if it has determinantal complexity at most $s$, then it can be computed by a homogeneous algebraic branching program (ABP) of size at most $O(d^5s)$. Moreover, we show that for $\textit{most}$ homogeneous polynomials, the width of the resulting homogeneous ABP is just $s-1$ and the size is at most $O(ds)$. Thus, for constant degree homogeneous polynomials, their determinantal complexity and ABP complexity are within a constant factor of each other and hence, a super-linear lower bound for ABPs for any constant degree polynomial implies a super-linear lower bound on determinantal complexity; this relates two open problems of great interest in algebraic complexity. As of now, super-linear lower bounds for ABPs are known only for polynomials of growing degree, and for determinantal complexity the best lower bounds are larger than the number of variables only by a constant factor. While determinantal complexity and ABP complexity are classically known to be polynomially equivalent, the standard transformation from the former to the latter incurs a polynomial blow up in size in the process, and thus, it was unclear if a super-linear lower bound for ABPs implies a super-linear lower bound on determinantal complexity. In particular, a size preserving transformation from determinantal complexity to ABPs does not appear to have been known prior to this work, even for constant degree polynomials.